We are concerned with the engineering of user-defined band gaps in periodic structures. Using a hollow unit cell as a prototype problem, we describe a systematic framework for the shape optimization of unit cells by suitably adapting the framework we had developed earlier for material optimization of unit cells. In contrast to most developments to date, we drive the optimization using a group-velocity-based objective functional, constrained only by the unit cell’s dispersive behavior, where the latter is expressed in terms of the associated quadratic eigenvalue problem: perfect gaps result at user-defined frequency ranges. The framework can be used not only to engineer omni-, uni-, or multidirectional gaps but could accommodate other wave-control objectives, as long as such goals can be expressed in terms of the group velocity. To exemplify the development, we parameterize the hollow cell’s trial cavity shape using newly developed specialized elements with C1 continuity along the cavity boundary and seek to uncover its shape parameters by forcing the vanishing of the group velocity at discrete frequencies spanning the target band gap, subject to the underlying dispersion relation. We use the unit cell’s Floquet–Bloch eigenvalue problem to describe the dispersion constraint and appeal to the Hellmann–Feynman theorem to relate the group velocity to the Floquet–Bloch eigenpair. In this manner, a user-defined band gap objective is directly connected to the design shape variables, while, and rather importantly, the choice of a group-velocity-based objective functional ensures the synergy of Bragg and local resonance behavior in spanning the target band gap. We resort to a gradient-based scheme to resolve numerically the first-order optimality conditions while appealing to the Reynolds transport theorem to address the evolving shapes and changing geometric domains during inversion iterations. Numerical experiments in two dimensions confirm that the shape-optimized unit cells attain user-defined, omnidirectional band gaps. The developed design process is capable of tackling complex metamaterial design problems and yield metamaterial assemblies that enable applications in noise mitigation, seismic shielding, and cloaking.
Khazi et al. (Mon,) studied this question.